91Ó°ÊÓ

For the following figure, state which congruence method can be used to prove the triangle congruent. If no method applies, say none.

Copy and complete the proof.

The statements in Exercise 2 might be used as statements in a proof but they are given out of order. Find an appropriate order for the statements. (There may be more than one correct order.)

2. Given:$\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{M}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{B}\mathbf{M}}\mathbf{,}\stackrel{\mathbf{¯}}{\mathbf{T}\mathbf{M}}\mathbf{âŠ\yen }\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{B}}$

Prove:$\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{T}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{B}\mathbf{T}}$

$\begin{array}{l}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{M}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{B}\mathbf{M}}\\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{Δ}\mathbf{A}\mathbf{M}\mathbf{T}\mathbf{â‰\dots }\mathbf{Δ}\mathbf{B}\mathbf{M}\mathbf{T}\\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{∠}\mathbf{\text{1}}\mathbf{â‰\dots }\mathbf{∠}\mathbf{\text{2}}\\ \mathbf{\left(}\mathbf{d}\mathbf{\right)}\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{T}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{B}\mathbf{T}}\\ \mathbf{\left(}\mathbf{e}\mathbf{\right)}\stackrel{\mathbf{¯}}{\mathbf{T}\mathbf{M}}\mathbf{âŠ\yen }\stackrel{\mathbf{¯}}{\mathbf{A}\mathbf{B}}\\ \mathbf{\left(}\mathbf{f}\mathbf{\right)}\stackrel{\mathbf{¯}}{\mathbf{T}\mathbf{M}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{T}\mathbf{M}}\end{array}$

In Exercises 1-6 you are given a diagram that is marked with given information. Give the reason for each key step of the proof.

Prove:$\stackrel{¯}{\mathrm{BE}}â‰\dots \stackrel{¯}{\mathrm{DF}}$

Key steps of proof:

a. width="112" height="20" role="math">ΔABC≅ΔCDA

b.$∠\mathbf{1}â‰\dots ∠\mathbf{2}$

c.$\mathrm{Δ}\mathbf{ABE}â‰\dots \mathrm{Δ}\mathbf{CDF}$

d.$\stackrel{¯}{\mathrm{BE}}â‰\dots \stackrel{¯}{\mathrm{DF}}$

Describe your plan for proving the following.

2. Given: $\stackrel{\mathbf{¯}}{\mathbf{P}\mathbf{R}}$bisects $\mathbf{∠}\mathit{Q}\mathit{P}\mathit{S}$and $\mathbf{∠}\mathit{Q}\mathit{R}\mathit{S}$Prove: $\stackrel{\mathbf{¯}}{\mathbf{R}\mathbf{Q}}\mathbf{â‰\dots }\stackrel{\mathbf{¯}}{\mathbf{R}\mathbf{S}}$

In the following figure, the two-triangle shown are congruent. Then complete the following statement.

$\mathrm{Δ}PQRâ‰\dots \mathrm{Δ}\_\_\_$

Complete.

If$\stackrel{\mathbf{¯}}{\mathbf{R}\mathbf{K}}\mathbf{âŠ\yen }\stackrel{\mathbf{¯}}{\mathbf{S}\mathbf{T}}$,then$\stackrel{\mathbf{¯}}{\mathbf{R}\mathbf{K}}$ is called$\mathit{a}\left(n\right)\underset{\mathbf{¯}}{\mathbf{?}}$ of$\mathbf{Δ}\mathit{R}\mathit{S}\mathit{T}$

Given: $\mathit{Δ}\mathit{K}\mathit{O}\mathit{P}\mathbf{â‰\dots }\mathit{Δ}\mathit{M}\mathit{A}\mathit{T}$

Name three pairs of corresponding sides.

For the following figure, find the value of$\mathit{x}$.

For the following figure, if $\mathrm{Δ}BOC$is isosceles with $\stackrel{¯}{OB}â‰\dots \stackrel{¯}{OC}$then$∠\_\_â‰\dots ∠\_\_$.

$\stackrel{\mathit{¯}}{\mathit{\text{OR}}}$is a common side of two congruent quadrilaterals.

Complete: quad.$\mathit{\text{NERO}}â‰\dots$quad$\underset{¯}{?}$.